# Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions - Mathematics > Probability

Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions - Mathematics > Probability - Download this document for free, or read online. Document in PDF available to download.

Abstract: We consider a process $X t {t\in0,T}$ given by the SDE $dX t = \alphabtX t dt + \sigmat dB t$, $t\in0,T$, with initial condition $X 0=0$,where $T\in0,\infty$, $\alpha\in R$, $B t {t\in0,T}$ is a standard Wienerprocess, $b:0,T\to R\setminus\{0\}$ and $\sigma:0,T\to0,\infty$ arecontinuously differentiable functions. Assuming that $b$ and $\sigma$ satisfy acertain differential equation we derive an explicit formula for the jointLaplace transform of $\int 0^t\frac{bs^2}{\sigmas^2}X s^2 ds$ and$X t^2$ for all $t\in0,T$. As an application, we study asymptotic behaviorof the maximum likelihood estimator of $\alpha$ for $\sign\alpha-K=\signK$,$K e0$, and for $\alpha=K$, $K e0$. As an example, we examine the so-called$\alpha$-Wiener bridges given by SDE $dX t = -\frac{\alpha}{T-t}X t dt + dB t$,$t\in0,T$, with initial condition $X 0=0$.

Author: ** Matyas Barczy, Gyula Pap**

Source: https://arxiv.org/